Stephen G. CECCHETTI Kermit L. SCHOENHOLTZ Bonds, Bond Prices, and the Determination of Interest Rates Copyright © 2011 by The McGraw-Hill Companies, Inc.

Stephen G. CECCHETTI • Kermit L. SCHOENHOLTZ

Bonds, Bond Prices, and the Determination of Interest Rates

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Chapter Six

6-2

Introduction• Car loans, home mortgages, and even credit

card balances all create a loan from a financial intermediary - just like government and corporate bonds.

• Virtually any financial arrangement involving the current transfer of resources from a lender to a borrower, with a transfer back in the future, is a form of a bond.

• This free flow of resources through bond markets is essential to a well functioning economy.

6-3

Bond Prices

• A standard bond specifies the fixed amounts to be paid and the exact dates of the payments.

How much should you be willing to pay for a bond?

• That depends on the bond characteristics.• We will examine four basic types.

6-4

Bond Prices1. Zero-coupon or discount bond

• Promise a single payment on a future date• Example: Treasury bill

2. Fixed-payment loan• Sequence of fixed payments• Example: Mortgage or car loan

3. Coupon bond• periodic interest payments + principal repayment at maturity • Example: U.S. Treasury Bonds and most corporate bonds

4. Consol• periodic interest payments forever, principal never repaid• Example: U.K. government has some outstanding

6-5

ni)1(

100$

Zero-Coupon Bonds

• U.S. Treasury bills (T-bills) are the most straightforward type of bond.• Each T-bill represents a promise by the U.S.

government to pay $100 on a fixed future date.• No coupon payments - zero-coupon bonds• Also called pure discount bonds (or discount bonds)

since the price is less than face value - they sell at a discount.

• Price of $100 face value zero-coupon bond

6-6

Assume i = 5%

Price of a One-Year Treasury Bill

Price of a Six-Month Treasury Bill

59.97$)05.01(

1002/1

24.95$)05.01(

100

Zero-Coupon Bonds

6-7

Zero-Coupon Bonds

• For a zero-coupon bond, the relationship between the price and the interest rate is the same as we saw on present value calculations.

• When the price moves, the interest rate moves with it, in the opposite direction.

• We can compute the interest rate from the price using the present value formula.

The price of a one-year T-bill is $95.

i = ($100/$95) - 1 = 0.0526 = 5.26%

6-8

FixedPayment

(1 i)

FixedPayment

(1 i)2 FixedPayment

(1 i)n

Fixed-Payment Loans• Home mortgages and car loans are fixed-payment

loans.• They promise a fixed number of equal payments at regular

intervals.• Amortized loans - the borrower pays off part of the principal

along with the interest for the life of the loan.

• Value of a Fixed Payment Loan =

• The sum of the present value of the payments.

6-9

• The issuer of a coupon bond promises to make a series of periodic interest payments (coupon payments), plus a principal payment at maturity.

Price of Coupon Bond =

nCB i

entCouponPaym

i

entCouponPaym

i

entCouponPaymP

)1(......

)1()1( 21

ni

FaceValue

)1(

Coupon Bonds

6-10

i

PaymentCoupon Yearly PConsol

Consols

• Consols or perpetuities, are like coupon bonds whose payments last forever.

• The borrower pays only interest, never repaying the principal.

• The U.S. government sold consols once in 1900, but the Treasury has bought them all back.

• The price of a consol is the present value of all future interest payments.

6-11

Bond Yields• We know how to calculate bond prices given an

interest rate.• We also need to be able to go in the other direction.

• Calculate the return to an investment, implicit in the bond’s price.

• We will combine information about the promised payments with the price to obtain the yield:• A measure of the cost of borrowing and the reward for

lending.• We will use the terms yield and interest rate interchangeably.

6-12

• The most useful measure of the return on holding a bond is called the yield to maturity:• The yield bondholders receive if they hold the bond

to its maturity when the final principal payment is made.

Price of 1yr 5% Coupon Bond =

• The value of i that solves the equation is the yield to maturity.

)1(

100$

)1(

5$

ii

Yield to Maturity

6-13

Yield to Maturity• If Bond Price = $100,

Yield to maturity = the coupon rate

• If Bond Price > $100, Yield to maturity < the coupon rate

• If Bond Price < $100, Yield to maturity > the coupon rate

Bond Price Yield to maturity

6-14

Yield to Maturity

• This relationship should make sense.• If you pay $95 for a $100 face value bond, you

will receive both the interest payments and the increase in value from $95 to $100.

• This rise in value is referred to as a capital gain and is part of the return on your investment.

• When the price of a bond is higher than face value, the bondholder incurs a capital loss.

6-15

Paid Price

Payment CouponYearly YieldCurrent

Current Yield

• Current yield is the measure of the proceeds the bondholder receives for making a loan.

• The current yield measures that part of the return from buying the bond that arises solely from the coupon payments.

• If the price is below par, the current yield will be below the yield to maturity.

6-16

0505.099

5

99$)1(

100$

)1(

5$

ii

Current Yield

Example: 1 year, 5% coupon bond selling for $99 Current Yield = , or 5.05%

Yield to maturity for this bond is 6.06 percent found as the solution to:

6-17

Current Yield

• We can see the relationship between the current yield and the coupon rate.

• Table 6.1 summarizes these relationships.

6-18

Holding Period Returns

• The holding period return is the return to holding a bond and selling it before maturity.

• The holding period return can differ from the yield to maturity.

6-19

Examples:• 10 year bond• 6% coupon rate• Purchase at face value, $100• Hold for one year and then sell it


6-20

What if the interest rate falls to 5%?

1yr Holding Period Return =

1-yr Holding Period Return = 13.11%

1311.0100$

11.13$

100$

100$11.107$

100$

6$


6-21

0052.0100$

52.0$

100$

100$48.93$

100$

6$


What if the interest rate rises to 7%?

1yr Holding Period Return =

1-yr Holding Period Return = 0.52%

6-22

Bond the of Price

Bond the of Price in Change

Paid Price

Payment CouponYearly


• The one-year holding period return is the sum of the yearly coupon payment divided by the price paid for the bond and the change in the price divided by the price paid.

= Current Yield + Capital Gain (as a %)

6-23


• Whenever the price of a bond changes, there is a capital gain or loss.

• The greater the price change, the more important that part of the holding period return becomes.

• The longer the term of the bond, the greater those price movements and associated risk can be.

6-24

The Bond Market and the Determination of Interest

Rates• How are bond prices determined and why do

they change?• We must look at bond supply, bond demand,

and equilibrium prices.• First we will restrict discussion to the quantity of

bonds outstanding - stock of bonds.• Secondly, we will talk about bond prices rather than

interest rates.• Finally, we will consider the market for a one-year

zero-coupon bond with a face value of $100.

6-25

• Assume an investor is planning to purchase a one-year bond and hold until maturity.• They have a one-year investment horizon.• The holding period return equals the bond’s yield to

maturity - both determined directly from price.• The present value formula shows the

relationship between price and yield as:

P

P100$ior

i1

100$P

The Bond Market and the Determination of Interest

Rates

6-26

Bond Supply, Bond Demand and Equilibrium in the Bond

Market• Supply and demand determine bond prices

(and bond yields).• The bond supply curve is the relationship

between the price and the quantity of bonds people are willing to sell, all else equal.

• The higher the price of a bond, the larger the quantity supplied.• The bond supply curve slopes upward.• Why?

6-27


Market1. Investor:

• The higher the price, the more tempting it is to sell a bond they currently hold.

2. Company seeking to finance a project:• The higher the price at which they can sell bonds,

the cheaper it is to borrow.• For a $100 one-year zero-coupon bond, the

supply will be higher at $95 than it will be at $90, all other things being equal.

6-28


Market• The bond demand curve is the relationship

between the price and the quantity of bonds that investors demand, all else equal.

• The price of bonds is inversely related to the yield, the demand curve implies that the higher the demand for bonds, the higher the yield.

• The bond demand curve slopes downward.• Why?

6-29

• Investors:• The lower the price bondholders must pay for a

fixed-dollar payment on a future date, the more likely they are to buy a bond.

• For a $100 one-year zero-coupon bond, the demand will be higher at $90 than it will be at $95, all other things being equal.


Market

6-30

Equilibrium is the point at which supply equals demand.


Market

6-31


Market• What if bond prices are above equilibrium?

• Quantity supplied > quantity demanded.• With excess supply, supplier cannot sell bonds they

want at current price.• Suppliers begin to cut the price.• Excess supply puts downward pressure on the price

until supply equals demand.

6-32


Market• What if the price is below equilibrium?

• Those who wish to buy bonds cannot get what they want at that price.

• They start bidding up the price.• Excess demand puts upward pressure on the price

until supply equals demand.

6-33


Market• To understand bond prices, we really need to

understand what determines supply and demand.

• Remember the distinction between moving along a curve (change in quantity supplied or demanded) versus a shift in the curve (change in supply or demand).

• A shift in either supply or demand changes the price of bonds, so it changes yields as well.

6-34

Factors That Shift Bond Supply

1. Changes in Government Borrowing• Any increase in the government’s borrowing

needs increases the quantity of bonds outstanding, shifting the bond supply curve to the right.

2. Change in General Business Conditions• As business conditions improve, the bond supply

curve shifts to the right.

3. Changes in Expected Inflation• When expected inflation rises, the cost of

borrowing falls, shifting the bond supply curve to the right.

6-35

When borrowers’ desire for funds increases, the supply curve shifts to the right.

This lowers bond prices, raising interest rates.

Factors That Shift Bond Supply

6-36

6-37

Factors That Shift Bond Demand

1. Wealth• Increases in wealth shift the demand for bonds to the right.

2. Expected Inflation• Declining inflation means promised payments have higher

value - bond demand shifts right.

3. Expected Returns and Expected Interest Rates• If the return on bonds rises relative to the return on

alternative investments, bond demand will shift right.• When interest rates are expected to fall, price prices are

expected to rise shifting bond demand to the right.

6-38


4. Risk Relative to Alternatives• If bonds become less risky relative to alternative

investments, demand for bonds shifts right.

5. Liquidity Relative to Alternatives• Investors like liquidity: the more liquid the bond,

the higher the demand.• If bonds become less risky relative to alternative

investments, demand for bonds shifts right.

6-39

When bonds become more attractive for investors, the demand curve shifts to the right.

This raises bond prices, lowering interest rates.


6-40


6-41


6-42

Understanding Changes in Equilibrium Bond Prices and

Interest Rates• Recall that expected inflation affects both bond

supply and bond demand.• Increasing expected inflation

• Reduces the real cost of borrowing shifting bond supply to the right.

• But, lowers real return on lending, shifting bond demand to the left.

6-43

These two effects reinforce each other.• Lower bond prices• Raise interest rates

D1

S1

Pric

e pe

r B

ond

Quantity of Bonds

D0

S0

E0

E1

1) Shift S right

2) Shift D right

However, graphically unknown effect on quantity of bonds.


Interest Rates

6-44


Interest Rates• A business cycle downturn:

• Reduces business investment opportunities shifting bond supply to the left.

• Reduces wealth, shifting bond demand to the left also.

• Although both shifts could give an ambiguous answer in terms of bond price, theory is not ambiguous.• In recessions, interest rates tend to fall meaning that

bond prices should rise.

6-45

We know price increases so supply shifts more than demand.

S0

Pric

e pe

r B

ond

Quantity of Bonds

D0

E0


Interest Rates1) Shift S left

2) Shift D left

D1

E1

S1

6-46

Why Bonds are Risky

Bondholders face three major risks.

1. Default risk is the chance that the bond’s issuer may fail to make the promised payment.

2. Inflation risk means investors can’t be sure of what the real value of payments will be.

3. Interest-rate risk arises from a bond-holder’s investment horizon, which may be shorter than the maturity date of the bond.

6-47

Why Bonds are Risky

• Risk arises because an investment has many possible payoffs during the holding horizon.

• We need to look at the risk the bondholder faces, what are the possible payoffs, and how likely each is to occur.

• We will compare the risk on a bond’s return relative to the risk-free rate.

6-48

Default Risk• Although there is little to no default risk with

U.S. Treasury bonds, there is with corporate bonds.

• In determining what happens to bond prices when we consider default risk, we can look at an example of a corporate bond.

• Assume the one-year risk-free interest rate is 5 percent.

• A company has issued a 5 percent coupon bond with a face value of $100.

6-49

Default Risk

• If this bond was risk-free, the price of the bond would be the present value of the $105 payment.Price of risk free bond = ($100 + $5)/$1.05 = $100

• Suppose, however, there is a 10% probability that the company will go bankrupt before paying back the loan.• Assume the outcome is either $105 or $0.• Expected value equals $94.50

Price of bond = $94.50/1.05 = $90

6-50

Default Risk• If the price of the bond is $90, what yield to

maturity does this price imply?Promised yield on bond = $105/$90 - 1 = 0.1667

• Since the default risk premium is the promised yield to maturity minus the risk-free rate:

= 16.67 percent - 5 percent = 11.67 percent.• Any risk premium will drive the price below

$90 and push the yield to maturity above 16.67 percent.

• The higher the default risk, the higher the yield.

6-51

• Inflation-indexed bonds are structured so that the government promises to pay you a fixed interest rate plus the change in the consumer price index (CPI).

• The U.S. Treasury sells two types of bonds that are guaranteed to beat inflation.

1. Series I savings bonds

2. Treasury Inflation Protection Securities (TIPS)

6-52

Inflation Risk

• With few exceptions, bonds promise to make fixed-dollar payments.

• Remember that we care about the purchasing power of our money, not the number of dollars.• This means bondholders care about the real interest

rate.

• How does inflation risk affect the interest rate?

6-53

Inflation Risk

• Think of the interest rate having three components:

1. The real interest rate

2. Expected inflation, and

3. Compensation for inflation risk.

• Example:• Real interest rate is 3 percent.• Inflation could be either 1 percent or 3 percent.• Expected inflation is 2 percent, with a standard

deviation of 1.0 percent.

6-54

Inflation Risk

• Nominal interest rate should equal• 3 percent real interest rate +• 2 percent expected inflation +• Compensation for inflation risk

• The greater the inflation risk, the larger the compensation for it.

6-55

Interest-Rate Risk

• Interest-rate risk arises from the fact that investors don’t know the holding period return of a long-term bond.• The longer the term of the bond, the larger the price

change for a given change in the interest rate.

• For investors with holding periods shorter than the maturity of the bond, the potential for a change in interest rates creates risk.• The more likely the interest rates are to change

during the bondholder’s investment horizon, the larger the risk of holding a bond.

6-56

• When the boom in house prices ended, holders of the private-label securities suffered large losses.

• In the resulting financial crisis in 2007, private securitization of mortgages collapsed.

• Although it has yet to rebound, some underwriters were seeing an opportunity to revive it.

Stephen G. CECCHETTI • Kermit L. SCHOENHOLTZ

Bonds, Bond Prices, and the Determination of Interest Rates

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

End ofChapter Six

Stephen G. CECCHETTI Kermit L. SCHOENHOLTZ Bonds, Bond Prices, and the Determination of Interest Rates Copyright © 2011 by The McGraw-Hill Companies, Inc.

Documents

price of coupon bond

coupon rate bond price

bond prices

bonds price

coupon bond periodic

maturity slide

bond markets

standard bond